# Math Help - Two Dimensional Discrete Dynamical System

1. ## Two Dimensional Discrete Dynamical System

Hi, I have the following dynamical system,

$x_{n+1}={x_n}^2-{y_n}^2+a$
$y_{n+1}=2x_ny_n$

Where a is real.

And I am asked to consider the set of points on a circle of radius r and centre origin and show that they are mapped to another circle under one iteration.

How do I go about this? I know the equation for the circle is of course $x^2+y^2 = r^2$. Can someone please give me a clue how to start?

Thanks,
Katy

2. ## Re: Two Dimensional Discrete Dynamical System

Let
$x_0^2+y_0^2=r^2$ be the initial circle to be mapped by the system.
Calculate $x_1,\text{ and } y_1$, then you can find that $x_1^2+y_1^2= \dots$, i.e. a circle with the origin is translated along x axis.

There you go, Katy